A model for fluctuations of the spatial mean in a turbulent channel - - PowerPoint PPT Presentation

A model for fluctuations of the spatial mean in a turbulent channel flow: a window on physics beyond the periodic box.

Paolo Luchini DIIN, Universit` a di Salerno, Italy Maurizio Quadrio Department of Aerospace Sciences and Technologies, Politecnico di Milano, Italy EDRFCM – Bad Herrenalb March 29th, 2019

A DNS numericist lives in a periodic box

A DNS numericist lives in a periodic box

and is obliged to choose between assigning a flow rate or a pressure gradient.

A DNS numericist lives in a periodic box

and is obliged to choose between assigning a flow rate or a pressure gradient. In reality neither the one nor the other is constant. Does the box have a window upon reality?

Background

• In Direct Numerical Simulations of drag reduction, when the drag

reducing device is switched on, a temporal transient occurs which needs to be discarded if one wants to obtain reliable mean values.

• Empirically, the transient is much longer when the simulation is

performed at constant pressure gradient (CPG) than at constant flow rate (CFR).

• This difference, while favouring CFR for practical reasons, revives the
• ld question of which between CPG and CFR conditions (both artificial

to some extent) is closer to reality.

• A third alternative, constant power input (CPI) was introduced by

Hasegawa et al. (2014) as a possibly more physical compromise.

Background

• In Direct Numerical Simulations of drag reduction, when the drag

reducing device is switched on, a temporal transient occurs which needs to be discarded if one wants to obtain reliable mean values.

• Empirically, the transient is much longer when the simulation is

performed at constant pressure gradient (CPG) than at constant flow rate (CFR).

• This difference, while favouring CFR for practical reasons, revives the
• ld question of which between CPG and CFR conditions (both artificial

to some extent) is closer to reality.

• A third alternative, constant power input (CPI) was introduced by

Hasegawa et al. (2014) as a possibly more physical compromise. Subject of this presentation will be a physical interpretation of the

• ccurrence of different transients in CPG and CFR, and a simple predictive

model for the behaviour of more general conditions such as CPI.

Observed transients

Left: CFR simulation from Quadrio and Ricco, 2003. Right: CPG simulation from Ricco et al., 2012.

Transients versus spectra

Can we obtain information about the transients of a system by just looking at the frequency spectrum of its natural fluctuations?

Transients versus spectra

Can we obtain information about the transients of a system by just looking at the frequency spectrum of its natural fluctuations? The answer is yes for a linear system driven by white noise: dx dt + Ax = yδ(t) ⇓ x = H(t)y with H(t) = exp(−At) dx dt + Ax = n ⇓ Sxx(ω) = F(H)F(HT)

Transients versus spectra

Can we obtain information about the transients of a system by just looking at the frequency spectrum of its natural fluctuations? The answer is yes for a linear system driven by white noise: dx dt + Ax = yδ(t) ⇓ x = H(t)y with H(t) = exp(−At) dx dt + Ax = n ⇓ Sxx(ω) = F(H)F(HT) That a similar relationship applies to slow enough (macroscopic) transients

• f a nonlinear microscopic system is the foundation of the

fluctuation-dissipation theorem of nonequilibrium thermodynamics.

Separation of microscopic and macroscopic scales in a turbulent flow

Essential to nonequilibrium thermodynamics is the scale separation between microscopic and macroscopic phenomena. In turbulence there is no scale separation. Or, wait a minute. . .

Separation of microscopic and macroscopic scales in a turbulent flow

Essential to nonequilibrium thermodynamics is the scale separation between microscopic and macroscopic phenomena. In turbulence there is no scale separation. Or, wait a minute. . . There is none up to the diameter (or wall half distance h in 2D) of a duct. But the diameter is geometrically separated from the longitudinal (virtually infinite) scale of length. What happens on a length much larger than h, or

• n a time much larger than h/uτ, actually is scale-separated.

With respect to the slowest time scales (the transients), the rest of the turbulence is, to a first approximation, white noise, just as in statistical physics. Can we identify the underlying “macroscopic” system from the spectra?

Fluctuations of averaged quantities

In a DNS of channel flow, the most obvious quantities of interest are the wall shear stress τw and the flow rate represented by the bulk velocity U. These are average quantities, which fluctuate only as an effect of the finite space and time samples involved in their averaging. Before proceeding we have to make sure that the fluctuations of average quantities have a physical and not just a numerical meaning.

Frequency spectra of sample means

¯ f (t) = 1 N

N

• i=1

fi(t) ¯ f

• = fi

¯ f (t) − ¯ f ¯ f (t + τ) − ¯ f

• = 1

N2

N

• i=1

(fi(t) − fi) (fi(t + τ) − fi) = 1 N (fi(t) − fi) (fi(t + τ) − fi) S¯

f ¯ f (ω) = 1

N Sfifi(ω)

• The ensemble mean of N independent samples has the same statistical

mean but N−1/2 times the fluctuation as an individual sample. However,

• the spectrum of fluctuations remains proportional to itself, and its

characteristic frequency and time scale are the same.

Frequency spectra of spatial means

¯ f (t) = 1 L L f (t, x)dx ¯ f (t)

• = f (t, x)

¯ f (t)¯ f (t + τ)

• = 1

L2 L L f (t, x1)f (t + τ, x2) dx1dx2 ≃ 1 L lim

L→∞

L

−L

f (t, 0)f (t + τ, ξ) dξ S¯

f ¯ f (ω) = 1

LSff (ω, 0)

• The spatial mean over an interval (or a box in multiple dimensions) of

size L has the same statistical mean but L−1/2 times the fluctuation as an instantaneous and localized value. However,

• the spectrum of fluctuations is the same, and so is its characteristic

frequency and time scale.

The equation for the spatial mean flow rate

a.k.a. (0,0) spatial Fourier mode

ρ∂U ∂t + ∂p ∂x + τw2 − τw1 2h = 0 A general external-forcing condition can be written as a linear combination

• f pressure gradient and bulk velocity:

∂p ∂x − ZGU = VG. Coefficient ZG is dimensionally a (possibly complex) generator impedance.

A “noisy resistor” model of the wall shear stress

fluctuation n(t) τ ′

w = R 2hU′

• White-noise assumption follows from scale separation:

τw = cf ρU2 2 + n where n is a white noise, or just uncorrelated at large enough scales.

• “Macroscopic” (slow) fluctuations can be linearized:

τ ′

w ≃ R 2hU′,

where R = (2h)−1 dτw dU = 2¯ τw 2h ¯ U

• 1 + κ−1

cf /2 −1 ≃ ¯ τw h ¯ U .

Electrical analogy

VG ZG bulk vel. U pressure gradient fluctuation n1(t) τ ′

w1

2h = RU′ ρ∂U ∂t fluctuation n2(t) τ ′

w2

2h = RU′ Typical first-order LR low-pass filter with noisy resistors. All components are independent of the computational-box size!

Orders of magnitude

• Typical time constant of the τw shear-stress fluctuation: ∼ h/uτ, based
• n a typical velocity uτ and size h of the largest vortices. Typical

ω ≃ 2πuτ/h.

• Differential resistance easily estimated as the derivative of Prandtl’s law:

R = (2h)−1 dτw dU = 2¯ τw 2h ¯ U

• 1 + κ−1

cf /2 −1 ≃ ¯ τw h ¯ U . The equivalent circuit (for ZG = 0, CPG) is then a classical first-order low-pass filter with time constant

VG ZG bulk vel. U pressure gradient fluctuation n1(t) τ ′

w1

2h = RU′ ρ∂U ∂t fluctuation n2(t) τ ′

w2

2h = RU′

L 2R = ρUh 2τw = 1 2 U uτ h uτ . The key observation here is scale separation: the time constant of the RL filter is ∼ U/uτ times longer than the characteristic time of the shear-stress fluctuations.

Numerical test

Low-frequency end of the spectrum of CPG velocity fluctuations (which would not exist in the CFR case), compared with the spectrum produced by the equivalent circuit when forced by white noise or by the CFR fluctuations. Data taken from an existing channel-flow DNS database (Quadrio, Frohnapfel and Hasegawa, 2016), for CFR, CPG, and CPI cases at Reτ = 200. The datasets contain the time history of wall shear, bulk velocity and pressure gradient, sampled every 0.2 viscous time units, for a duration of 150,000 viscous time units. Hence each dataset contains 750,000 samples.

Spectrum of the temporal velocity fluctuations in CPG compared with the spectrum of the equivalent circuit.

10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-1 100 101 102 S(ω) ω h/uτ CPG U CFR τw

• equiv. circuit

Transfer function from CFR shear-stress fluctuations to CPG velocity fluctuations.

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 10-1 100 101 102 |H(ω)|2 ω h/uτ transfer function

• equiv. circuit

The U fluctuation is passively determined by the equivalent circuit and exerts no feedback on the τw fluctuation ⇒ It’s ok to neglect it! (CFR)

Generator-impedance representation of CPI

P = −pxU = const. δpx px = −δU U ⇓ ZG = −px U = τw hU = R L 2R + ZG = 2 3 L 2R

• The CPI transient is slightly (33%) shorter than the CPG transient.
• The fluctuations of bulk velocity and pressure gradient are of equal

relative amplitude.

An infinity-mirror generator impedance

Infinite 2D lattice of equal impedances: a classic exact solution in circuit theory.

Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z B Z A · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

=

Z A Z B

Sounds familiar? Added mass!

The Impedance Lattice forcing is a mirror condition

pressure gradient RU′ ρ∂U′ ∂t fluctuation 2n2(t) RU′ RU′ ρ∂U′ ∂t RU′

• Same time constant as CPG.
• p′

x = n2 = p′ x,CFR/

√ 2; U′ = U′

CPG/

√ 2. ⇓ The relative fluctuations of bulk velocity are much smaller than those of pressure gradient (contrary to CPI).

Equivalent circuits (fluctuations only)

CFR

pressure gradient bulk vel.= 0 fluctuation n1(t) RU′ ρ∂U′ ∂t fluctuation n2(t) RU′

CPG

pressure gradient= 0 bulk vel. U fluctuation n1(t) RU′ ρ∂U′ ∂t fluctuation n2(t) RU′

CPI

−px U bulk vel. U fluctuation n1(t) RU′ ρ∂U′ ∂t fluctuation n2(t) RU′

IL

pressure gradient RU′ ρ∂U′ ∂t fluctuation 2n2(t) RU′ RU′ ρ∂U′ ∂t RU′

Conclusions

Physical

• A “noisy resistor” model of wall shear stress is sufficient to represent

fluctuations (and transients) of the bulk velocity. Nonequilibrium thermodynamics is valid in the plan view of a 2D channel.

Numerical

• An Infinite-Lattice forcing condition offers a window on physics beyond

the periodic box. Incidentally this has the same time constant as CPG.

• In IL the U fluctuation is uτ/U times smaller than the px fluctuation.

CPI is suboptimal: it constrains them to be the same.

• The velocity fluctuation is passive: in old, well-tested CFR it can safely

be neglected in order to obtain a uτ/U times shorter artificial transient.

• A simple low-pass equivalent circuit gives you all 4 conditions.

Conclusions

Physical

• A “noisy resistor” model of wall shear stress is sufficient to represent

fluctuations (and transients) of the bulk velocity. Nonequilibrium thermodynamics is valid in the plan view of a 2D channel.

Numerical

• An Infinite-Lattice forcing condition offers a window on physics beyond

the periodic box. Incidentally this has the same time constant as CPG.

• In IL the U fluctuation is uτ/U times smaller than the px fluctuation.

CPI is suboptimal: it constrains them to be the same.

• The velocity fluctuation is passive: in old, well-tested CFR it can safely

be neglected in order to obtain a uτ/U times shorter artificial transient.

• A simple low-pass equivalent circuit gives you all 4 conditions.
• P.S. . . A 1D IL of circular pipes, as opposed to a 2D IL of channels, has

infinite impedance. The physical forcing for a pipe actually is CFR.